Hi,

I've problem with following convergence (in red shape). The second one term is tends to zero it is obvious but why the first one is tends to zero?

Thanks for any help.

http://img202.imageshack.us/img202/8928/lastkf.jpg

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- Apr 5th 2010, 09:56 PMArczi1984Convergence in compact Hausdorff space
Hi,

I've problem with following convergence (in red shape). The second one term is tends to zero it is obvious but why the first one is tends to zero?

Thanks for any help.

http://img202.imageshack.us/img202/8928/lastkf.jpg - Apr 5th 2010, 11:41 PMOpalg
You are told that $\displaystyle x_n\to x$ in the sup norm. This means that, given $\displaystyle \varepsilon>0$, there exists N such that $\displaystyle n\geqslant N\ \Rightarrow\ \|x_n-x\|_{\text{sup}} = \sup\{|x_n(t) - x(t)|:t\in X\} <\varepsilon$. Take $\displaystyle t=p_n$ to see that $\displaystyle n\geqslant N\ \Rightarrow\ |x_n(p_n) - x(p_n)|<\varepsilon$. Thus $\displaystyle |x_n(p_n) - x(p_n)|\to0$ as $\displaystyle n\to\infty$.